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Einstein published two theories of relativity
The Special Theory
For uniform motion 0a =
The General Theory
For non-uniform motion 0a ≠ .
First we will discuss The Special Theory
At the beginning of the century most believed
light was a wave.
Thus must have something that waves:
Sound has air
Water has water, etc.
Physicists proposed that for waves of light
something must wave.
They called it the “ether” for light
This ether then must fill the universe.
The earth moves through the universe so:
How fast are we traveling through the ether?
To answer this we will start with a race
between two boats (that run at exactly the same
speed) in a river.
The boat going downstream will have speed
c v+ The boat going upstream will have speed
The calculated time for round trip is
// 2 2
2L L Lct c v c v c v
= + = + − −
The speed for the boat going cross-stream will
have speed both ways
2 2c v−
Lt v c
Then the ratio of the two times will be
/ / 2
1 1 1c r o s s
t t v
Therefore we see that the boat that goes across
and back wins the race even though both boats
travel the same speed relative to the water.
The main point is the time to complete the
race is different for the two boats.
We can measure ratio of times for boats and
calculate the speed of river if we know the
speed of boats.
The Michelson-Morley Experiment
Diagram of Apparatus
Drawing of actual apparatus
We know the speed of light.
We want the speed of the earth through the
When light arrives at the eye it has traveled two
paths to reach the observer.
There will be interference – either
The resulting image will be a series of lines.
Since the direction of the ether flow is not
known the apparatus must be rotated.
First one than the other path will be parallel to
the flow of ether.
Therefore the interference lines should shift.
Michelson and Morley did the experiment very
carefully and did not find a shift.
The conclusion has to be that:
The Ether does not exist
the earth travels along with it.
Another experiment shows that we are
not moving with it.
Stellar Aberration is that experiment
While the light travels down the telescope the
telescope moves with the earth.
The telescope has to be tilted to keep the image
in the center.
If the ether (the substance that waves to cause
the propagation of light) moves with the scope
there would be no need to tilt the it.
Therefore we conclude the ether does not
The transformation equations before Einstein
x x v t
These are the Galilean Equations that allow
observers to compare observations in two
different frames moving relative to each other
with constant velocity.
Observer on ground and observer on railroad
car moving in x-direction.
The observer on the ground observes the birds
separated by distance 2 1x x− .
The distances are equal.
If an airplane flies over the railroad car
traveling in the x+ direction at a speed xu
measured by the observer on the ground what
will be the speed ( 'xu ) of the airplane measured
by the observer on the railroad car?
We can use the transformation equation for x
'x x vt= − and the equation for t
Differentiate and divide to get
'dx dx dvv t
dt dt dt = − −
' 0x xu u v= − −
if the velocity of the railroad car is constant.
If the observer on the ground measures the
velocity of the airplane as
then the person on the railroad car will measure
What if the person on the ground points a
flashlight in the x+ direction? What will be
the speed of light measured by the observer on
the railroad car?
x xu u v giving c c v
We must keep this result in mind as we discuss
Einstein’s postulates for the Special Theory of
1. Fundamental laws of physics are identical
for any two observers in uniform relative
2. The speed of light is independent of the
motion of the light source or observer.
These postulates cannot be satisfied using the
Galilean Equations, as we will see.
However Einstein found that the following
' ( ) ' '
' ( )
x x vt y y z z
vxt t c
= − = =
These are the Lorentz Transformation
Now consider the airplane flying over the
railroad car in the x-direction. What is the
speed of the airplane as measured by the
observer on the ground? What is the speed of
the airplane as measured by the observer on the
railroad car? We need to answer these
questions by using the Einstein-Lorenze
' ( )
' ( )
' ( )
' ( )
x x vt and
vxt t c
differentiate dx dx vdt and
vdxdt dt c
divide dx dx vdt
vdxdt dt c
divide by 'dt
dx vdx dt dxdt v dt
u vu vu c
Thus if an object (an airplane) flies over the
railroad car the observer on the ground will
measure the speed in the x direction as xu . The
observer on the car will find 'xu .
What about the speed of light when a flashlight
is pointed in the x-direction?
The observer on the ground points a flashlight
in the +x direction. What will be the speed of
light measured by the observer on the car?
2 ( ) /1 1
x c v c v c vu cvc v c v c
− − − = = = =
Both observers, even though they are moving
relative to each other, measure the same value
for the speed of light.
This is in agreement with the Second Postulate.
Read the section on Length Contraction in the
book. We will do it a little differently.
The observer on the moving railroad car has a
rod moving with him. He measures the length
of the rod to be
' ' 2 1 0x x L− =
Use the Lorentz equations to get
' 2 2 2
' 1 1 1
x x vt
x x vt
Then putting these in the equation
0 2 2 1 1
0 2 1 2 1
( ) ( )
( ) ( )
L x vt x vt
L x x v t t
= − − −
= − − −
If the observer on the ground measures the far
end and near end of the rod at the same time
1 2t t= Then
0 2 1( )L x x Lγ γ= − =
= and γ > 1
So the observer on the ground with the rod
moving past in the x direction measures the rod
to be shorter than what is measured by the
observer at rest relative to the rod and on the
Length Contraction is a prediction of the
Again we will find time dilation a different way